Here is the official Vatican announcement from papal spokesperson Ferderico Lombardi. I coped it this article posted at the National Public Radio website:
The brief meeting between Mrs. Kim Davis and Pope Francis at the Apostolic Nunciature in Washington, DC has continued to provoke comments and discussion. In order to contribute to an objective understanding of what transpired I am able to clarify the following points.
Pope Francis met with several dozen persons who had been invited by the Nunciature to greet him as he prepared to leave Washington for New York City. Such brief greetings occur on all papal visits and are due to the Pope's characteristic kindness and availability. The only real audience granted by the Pope at the Nunciature was with one of his former students and his family.
The Pope did not enter into the details of the situation of Mrs. Davis and his meeting with her should not be considered a form of support of her position in all of its particular and complex aspects.
We still don't know how Kim Davis was chosen for this meeting. Most commentators are saying that it was papal nuncio Archbishop Carlo Maria Vigano, an apparent supporter of Benedict XVI, who arranged the meeting.
If the Vatican statement above is true, then Mat Staver and the Liberty Counsel probably used this meeting for political purposes--spinning it in such a way so that they could drag Francis into the culture war over marriage. The Christian Right has placed political expediency over the whole truth.
Does the Pope and the Vatican regret meeting with Kim Davis? I don't know. They obviously regret the way this whole meeting has blown up in their faces. In that sense, I am sure some of the Pope's political handlers are upset with Vigano for letting Davis through the doors.
On the other hand, why can't the Pope's inclusive message of love and human connection be applied to Kim Davis, a person who, as misguided as she may be, has probably been through hell over the course of the last month? She has been mocked relentlessly for her beliefs on same-sex marriage, her failed marriages, her religion, her weight, and even her hairstyle. This seems like a perfect candidate for a visit from Francis--someone who is being bullied because of her beliefs. I can't help but believe that Jesus would have done the same thing.